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The Pennsylvania State University the Graduate School Eberly College of Science The Pennsylvania State University The Graduate School Eberly College of Science ALIE-ALGEBRAICAPPROACHTOTHE LOCALINDEXTHEOREMON COMPACTHOMOGENEOUSSPACES A Dissertation in Mathematics by Seunghun Hong c 2012 Seunghun Hong Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2012 The dissertation of Seunghun Hong was reviewed and approved∗ by the following: Nigel Higson Evan Pugh Professor of Mathematics Dissertation Advisor Chair of the Doctoral Committee John Roe Professor of Mathematics Ping Xu Professor of Mathematics Martin Bojowald Associate Professor of Physics Yuxi Zheng Professor of Mathematics Head of the Department of Mathematics ∗Signatures are on file in the Graduate School. ii ABSTRACT Using a K-theory point of view, R. Bott related the Atiyah-Singer in- dex theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This dissertation explains how to prove the local index theorem for compact homogenous spaces us- ing Lie algebra methods. The method follows in outline the proof of the local index theorem due to N. Berline and M. Vergne. But the use of B. Kostant’s cubic Dirac operator in place of the Rieman- nian Dirac operator leads to substantial simplifications. An impor- tant role is also played by the quantum Weil algebra of A. Alekseev and E. Meinrenken. iii CONTENTS LIST OF SYMBOLS vi ACKNOWLEDGMENTS viii 1 INTRODUCTION 1 2 REVIEW OF THE THEORY OF COMPACT LIE GROUPS 10 2.1 Analytical Aspects......................... 10 2.2 Algebraic Aspects.......................... 17 2.3 The Weyl Character Formula and the Spectrum of the Laplacian............... 26 3 THE VOLUME OF A COMPACT LIE GROUP 39 3.1 The Euler-Maclaurin Formula................. 40 3.2 An Example: SU(2) ......................... 42 3.3 The General Case.......................... 45 4 THE HEAT KERNEL OF THE LAPLACIAN ON A COMPACT LIE GROUP 50 4.1 The Duflo Isomorphism...................... 51 4.2 The Asymptotic Expansion of the Heat Kernel......................... 58 5 THE QUANTUM WEIL ALGEBRA 66 5.1 Clifford Algebras.......................... 66 5.2 The Quantum Weil Algebra................... 77 5.3 The Classical Weil Algebra.................... 82 5.4 The Relative Weil Algebra.................... 87 6 EQUIVARIANT DIFFERENTIAL OPERATORS AND THE QUANTUM WEIL ALGEBRA 92 6.1 Principal Bundles and Associated Vector Bundles.................. 92 6.2 Clifford Module Bundles and Spin Manifolds....... 99 6.3 Equivariant Differential Operators and the Relative Weil Algebra................... 104 7 THE ASYMPTOTIC EXPANSION OF THE HEAT KERNEL OF THE CUBIC DIRAC OPERATOR 109 7.1 The Convolution Kernel of a Generalized Laplacian.. 110 iv 7.2 The Asymptotic Expansion of the Heat Kernel of D= (g; k)2 .................. 112 8 THE LOCAL INDEX THEOREM ON COMPACT HOMOGENEOUS SPACES 118 8.1 Review of the Heat Kernel Proof of the Local Index Theorem................... 119 8.2 The Local Index Theorem on G=K .............. 124 9 THE DISTRIBUTIONAL INDEX OF THE CUBIC DIRAC OPERATOR 137 9.1 The Distributional Index of a Transversally Elliptic Operator.............. 138 9.2 The Distributional Index of D= (g; k) .............. 143 BIBLIOGRAPHY 149 v LISTOFSYMBOLS A, 134 D= (g; k), 88 exp∗, 53 Abas, 87 D= ±, 119 expCl, 74 Ak-bas, 88 D= g=k, 107 expV , 59 Ag, 87 D(A), 51 f ∼ g, 17 Ak, 87 D(G), 17 G fT , 33 Ahor, 87 D(G) , 58 Fr(F), 93 Ak-hor, 87 D(g), 58 FrO(F), 93 Aˆ , 123 D(g)G, 58 FrSO(F), 93 ad, 11 De, 52 Ad, 11 ∆G, 13 g¯, 77 Adf, 113 ∆g, 58 gC, 29 ads, 71 ∆ˆg, 84 gC,α, 29 Altλ, 34 δG, 53 gˆ, 78 Alt(µ), 142 δg, 53 Gb, 19 δij, 23 γ, 75 [ ; ]g, 77 D(g; k), 90 γg, 88 [ ; ] s, 69, 71 p diagW, 90 γ , 88 0 c, 100 D (M), 55 Γ(S), 4 Duf, 55 Γ 2E(G) Cµ, 141 , 37 Duf, 58 gr A C(G; C)fin, 24 , 56 C (G; C)fin, E, 125 HP, 94 1 24 e, 10 HpP, 94 ch, 123 eI, 68 ht, 15 χu, 18 e, 69 Cl(M), 99 I ind, 119 E0 (A), 52 l(M), 99 e Ind C 0 K, 140 E0(g), 53 Cl(V), 67 inds, 119 E0 (g)G, 55 Cl(n), 67 0 ι , 72 0 v k Ee(G), 53 Cl (V), 69 ιX, 77, 83 0 G Ee(G) , 55 Clk(V), 68 ", 52 j(X), 55, 130 Cl(V), 67 "G, 58 jk(X), 130 D= , 78, 81 "g, 58 jg=k(X), 130 vi ∗ κ, 12 P ×p iE, 93 TKM, 138 2 trV , 134 Lcl(G; C), 25 Q, 84 trg, 59 λ, 75 q, 69 p λ , 90, 125 uˇ, 18 Rf, 140 Λcoroot, 31 ± U(g), 20 Rf , 140 Λe, 28 Uk(g), 56 R(G), 35 Λg, 31 Rb(G), 37 [V], 35 ΛT , 28 ρ, 31 V(µ), 35 LX, 77, 83 ρg, 90 vol, 13 ν, 125 VP, 93 S, 102, 113 ± VpP, 93 Ω, 22 S , 75 Ω b g, 81 Sn, 67 W, 26 Ω (P; E) S(g) bas , 95 , 53 W(g), 82, 97 Ω(M; P × E) Sk(g) ν , 95 , 56 W(g; k), 90 O(tn) σ , 16 , 57 W(g), 77 O(t ) σ , 17 k, 56 W(g; k), 88 1 SpanR, 57 P0, 121 Str, 121 X, 77 PBW, 53 supp, 141 Xb, 78 Φ, 31 Xe, 10, 93 Φ+, 30 τ, 20, 51 Xee, 11 π, 125 T(g), 20 Ξ, 127 PSpin(M), 101 ΘV , 133 P ×G E, 92 ΘT , 133 Z(g), 23 vii ACKNOWLEDGMENTS It is a great pleasure to express my gratitude to the people who have helped me to finish this dissertation. I am grateful, foremost, to Professor Nigel Higson for introducing me to this beautiful area of mathematics. It is an honor for me to be a student of his. I am also grateful for the generous mental and financial support that he has provided. It is through Professor Loring Tu that I learned equivariant coho- mology and the Weil algebra, which has helped me tremendously in my research. As a matter of fact, had I not met him while I was a physics student at Tufts, I would not have switched my discipline and this dissertation would not be. I learned a great amount of mathematics (especially geometric functional analysis) from Professor John Roe through his lectures, talks, and books, and also through the various conversations we had. I thank Professor Ping Xu for his several thought-provoking com- ments on my research. Professor Martin Bojowald gave interesting feedbacks from the physics side. I learned the basics of K-theory and K-homology from Professor Paul Baum. To be around him has been joyful; I have always learned something when I was with him. When I had questions, he listened very carefully, and cheerfully shared his insights. Professor Alberto Bressan taught the course on second-order lin- ear partial differential equations when I took it. I am thankful for his taking time in discussing some of my questions surrounding the heat semigroup. I would like to thank Professor Adrian Ocneanu for the vibrant discussions we had on representation theory. As the Korean proverb says, “in a mizzle, one is not aware of his/her clothes getting wet”; so it is impossible for me to fully per- ceive all that I have absorbed from the intellectually stimulating and diverse environment promoted by the faculty of the Department of Mathematics at PENN STATE. But I do wish to mention that I owe much of my education to, in addition to the aforementioned profes- sors, Professor Anatole Katok, Professor Anna Mazzucato, Professor Kris Wysocki, Professor Nate Brown, Professor Steve Simpson, Pro- fessor Winnie Li, and Professor Yuri Zarkhin. I would like to also thank the staff members of the Department of Mathematics at PENN STATE. Especially Ms. Becky Halpenny; she has every reason to be called the “grad student mama.” viii Many people in the worldwide LATEX community have shared their work and ideas online; thanks to them, I was able to focus more on the content than the typesetting of this dissertation. It is impossible for me to fully appreciate all that I have received from my parents, Samsun Hong and Beomsik Roh; words can not fully express my gratitude. Additional thanks goes to my father for his comment on typography. I would also like to express my heartfelt gratitude to my parents- in-law, Sunggook Kim and Youngsook Jung, for their love and sup- port they have provided in various ways and forms. My greatest debt of love is to my wife Jeongwoon Kim — for her love, care, and patience. I must thank our daughter Zoyoung Hong too, for she has been an unceasing fount of cheer. A personal goal of mine was to complete the dissertation before she asks “When will you finish, daddy?” I think I made it. Finally, I owe sincere and earnest thankfulness to my church fam- ily, the congregation of Grace Presbyterian Church, for their support and prayers. And, of course, to the One who has listened and an- swered their prayers, Christ Jesus, the only wise God. ix Spectral point of view is the one which appears from experiments, when you study the universe, this is no fantasy. — A. Connes [49] INTRODUCTION1 ROM the sound waves of a drum to a quantum particle trapped F in a box, many fundamental physical systems can be studied by solving the eigenvalue problem of the Laplacian under Dirichlet boundary conditions: -∆φ = λφ, on U; (1.0.1) φ = 0; on @U; where U is a bounded open subset of a Euclidean space.
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